Integral error estimates-IV

INTEGRAL

ERROR

ESTIMATES-IV*

V. A. VINOKUROV Moscow (Received 12 September 1973; revised 11 May 1975)

THE CASE of a function, acting from metric space whose measure is matched in a natural way with its topology, into separable metric space is considered, and lower bounds are constructed for the error functional in integral norms, these bounds being independent of the method of solution.

Introduction The results obtained in [I] show that, for a wide class of the spaces used in analysis, the order of the error of an arbitrary method of solution is not greater than the order of the error of the trivial method, everywhere except for a set of the first category. On the other hand, the set of the first category may have a perfect measure. Jn Example 1 below we construct a continuous real function in a closed interval, in which the error can tend to zero faster than wcf, 6, x) in a set of perfect measure. We also obtain the inequality (1.3), representing a lower bound for the error integral for any method in terms of the properties of the initial function, and we mention some consequences of this bound. The assumptions and notation used in [l ,2] are retained throughout the present paper, while the following supplementary conditions are also introduced: it is assumed that the metric spaces X and Y are separable, and that a non-negative perfect u-finite measure p is given in space X, such that all open sets are measurable and all measurable sets are congruent to a set of the type F. (for brevity, we shah refer here to a set with the property S). These conditions on the connection of the measure with the topology are equivalent (see [3]) to identity of the class of functions which are regularizable almost everywhere, and the class of measurable functions. Further, f will denote throughout a kmeasurable function f: X + Y. In addition, arbitrary metric expansions XIxX, Y,xY and a mapping R from X1 into Yi, D (23) 3X will be fixed.

1. An integral inequality for the error

Let us first show that algorithms for the approximate solution exist for a continuous function, which converge to zero in a set of perfect measure more rapidly than w(f, 6, x). ExumpZe 1. L.etX=[O, 11, Y=[O, m); in the interval [0, 1 ] we define the continuous function fas the sum of the uniformly convergent series

*.Zh.vjkhisi.hfat.mat.Fiz., 16,3,549-566,1976. 3

4

V. A. Vinokurov

, are piecewise linear functions: fn(x) = 0 if Z+E(k6,-- A,, k&-l-A,), wheref,,n=1,2,... kb,]and [ k6,, fn (x), =2-“‘, if x = k6,, and are linearly interpolated in the intervals [/&,-A,, k&,+&J, where k=l, 2,. . . , 1/6,-l, 6, =2-2*“+n, A,=2”“‘.Let

then, by definition of fn(x), we have the relations

(1-l)

The local modulus of continuity c&, 6, x) differs little from the local modulus of continuity o (f,,, 6,, 2) .In fact, it follows from the properties o (gi-!-gz, 6, x) Go (gi, 6, 2) i-o (gz, 6, z), 0(-g, 6, r)=o(g, 6, x>thatlo(gl, 6: x)-o(g2, 6, x) J=%o(~,-g,, 6, z), i.e.,by(l.l),

“&a E in*

Ilfi’ll+ 2 llfill~2-n+“llfnll~ i-n+1

1. Denote by 17, the open set i/a,--!

G,={s~:: fn(sj>O}=

U

(k&,-A,,

k&+A,j,

k-i

and let M ,,= [ 0, I] \G,

then, for r&?,

and [x--x

denote its closed complement. Let the function

I<6

i.e., A (f, R,, 6n, s)~2-“+s1\fni1.

Integral error estimates

Thus, for every point ZEM,, and for n > 4, P+~ 4-2~n+3 .

A (f, Rn, 6n,xc>~ w (f, 6:,, 3) In short, this inequality holds for all n > 4 on the set

fi ilf,, but n= I\-

and hence, on the set of perfect measure

l.m

1 :i-+m

o

A (f, Rn, an,x> w(f,hl,x)

=

’

though the set M of the type F. is of the first category in [0, 11. 2. If we take n-1

R.(x)-~f.(x)+~llf.ll, i=i

as the regularizing sequence of mappings, we easily find that the quantities 1f(x) -R, (x’) ]for 1x-x’ I-G, satisfy

22

n-1

Ilfnll+ 2

Ilfi’l!G $ llfnll+2-n+3!!fni!.

Ilfill+L~

i=ny1

i--l

By the previous inequalities, for the local modulus of continuity we have

A (f, R,, a,,, x) ~ --1/2+2-“+3 1_24+J 0 (f, & x)

Vn34

and

which show that the constant H cannot be improved in the definition of the property of asymptotic estimation of [ I] . 3. Notice that the continuous function f constructed above is essentially asymmetric, in the sense that, if o+ (f, 6, x) = sup If (“+7)-f o==a4

(xl I,

o-(f,s,X)=o~~*If(x-y)-f(3)1,

6

V. A. Virwkurov

then, everywhere in [0, 1 ] except for a first category set,

For, if xE [ k6,+A,,

k6,+b&-A~I

o+ (f.,$,x)=o, i.e., o+(f,.

L/2,

x)
W4

o-(f.,$,x)=llf.ll,

x)Go(f-fn,

6”,

37)~2-n+311fnll, while

co-(f,6J2, x)~llfnll-2-n+311fnII. If we now put Qn =‘i;-i (k&,+A,, k=, the set

k6, + ;

- A,,),

9 o,, will be open and everywhere dense in [0, I], i.e., >.= .V

N=l

n=N

is the complement to a set of the first category, and if ZEQ

i7-g n+ca

A similar assertion is valid for O+ (f, b/2,

co-(f, w27 xl =

o+(f, &L/2,xl

o.

*

x) (co- ( f, 6,/2, x) ) -‘.

Our example shows that no deftite lower bound can be obtained.for the order of the error almost everywhere. In the present section we obtain a simple integral inequality for the error in the case of fixed 6 ; this inequality may be used to obtain asymptotic error estimates. Theorem 1 Given any p-measurable function fi X + Y, the metric expansions X,2X, function R’from X1 into Y,, D(R) XX, we have, for arbitrary 6 > 0, p > 1,

Y,xY

and the

(1.3)

a&J{ X

x+x:

J

p(x,,x,)

pP(f(r,),f(s3))dz3}~s,, C6

and if

aup CL{x+X: XIEX

p (Xi, z,) (6) “ql(6)
7

Integral error estimates

then

(1.4)

&~of: In accordance with the assumptions introduced at the beginning of this paper, the measure P is perfect, and the sets of measurable and of almost everywhere regularizable functions are identical; hence, by Theorem 8 of [2] , ACf,R, 6, x) is a measurable function of x for all 6 > 0. In the product of spaces XxX=X’ we consider the product of measures pXp=p.’ (see [4], p. 205). Every open set G of the space X2 will be &measurable, since, recalling that X is separable, X2 will also be separable, and will have a countable basis of sets of the type (U,, 17~)) where U1, 172are open spheres in X, i.e., G = ! (U,,, U,,). 1,---_1 We can thus say, first, that the function p (f(z,), f(~) ) is p2 -measurable, being the superposition of the continuous function p (x1, 52) and the &measurable function (f(z,) , f(sz) ), and second, that the open set La- { (21,zz) =x2: p (2*,22) (6) cxz is also &measurable. We take the fundamental inequality A(f,R,6,s,)+A(f,R,6,~z)~~(f(5,),f(52))

+i,~)=Lb,

raise it to the power p > 1, and, applying the numerical inequality (a+b) PG S’-’ (up+&‘), obtain 2p-‘[Ap(f,

R, 6, zi)+AP(f,

R, 6,~) lzpp(f(s,),

f(x,)).

Jn this last inequality, all the functions are /&measurable. We integrate it over the set Lg, while noting that the left-hand side is symmetric with respect to x1 and x2 :

By Tonelli’s theorem (see [4] , p. 213), the integrals can be written in the form of repeated integrals:

2pSAp(f,R,6,x,)k(E,,)ax,~

J { J pP(f(51)lf(x2))~~}~Z1, X

where E,,= {x+X: follows from (1.3).

p (q,

x2) (6)

-%

is an open sphere in space X, i.e., (1.3) is proved; and (1.4)

1. The constant in the inequality (1.4) is accurate. It is sufficient to take the following example: X={-i}U{+i}, Y= (-m, m), R (2) ~0, the function f is the identity embedding of X into Y, and the measure ccis equal to tl at the points -1 and +I. Then, Notes.

V. A. Vinokurov

8

and (1.4) becomes an exact equation. 2. Since a contraction of the measure /.Jonto a measurable subset EcX satisfies all the conditions of Theorem 1, the inequality (1.4) will remain true if X is replaced by E, and #(6) replaced by (PE(6)=SUp~{I?~E:p(5tr5*)
3. If, in Theorem 1, it is further required that the metric in X be almost convex, then the inequalities (1.3) and (1.4) can be strengthened, by replacing p (zr, 52)~6 throughout by p (51, 52)-=z%. It follows from the integral inequality (1.4) that the error may not be small at every point of a set of positive measure. Let us state an estimate of the measure. Theorem 2 If the measure /..ris finite, while the diameter of the set of values of the function f is 6(2’(f)} =M
vhO)~P(x)---EP)

y{x: A(f,RJ,x)>E}~p(X)

(5M)P-&P

(1.9

in the case of a metric space Yr , or

(1.6) in the case of a Hilbert space Yr,where pal,

J(p,6)=-

(f(4,f(4>hid%. JJ PP

1

zpq(6)

DO, O<&
X;,X~EX: p(x,,.x,)
PLoofi We go over from the mapping R to a &reasonable mapping R’ such that A (f, R’, 6,~) GA (1, R 6, x:)

VXEX;

this is possible, in accordance with Theorems 1 and 2 of [2]. Then, by Corollary 2 of [2], A(f,

inthe case of a metric

R',6,x)G3m(f, 6,~)+2a(f,26,x)G5M VXEX Yr , and by Note 3 of [2],

IIf

i.e.,

A(f,

-R’(d)

IKM

R',6,z)dkf

in the case of Hilbert space Yr . If E.={x:

A(f,

R',6,x)>E}, then

VXEX,

VXEX_

VxkD(R'),

Integral ffror estimates

s

A”(f, R’, 6, x) dx <

I

s

&pch+ (5M)9d~q(5M)*--&*)p(E,)+&*p(X) s X\E. ES

4

(1.7)

in the case of metric Y1, while

f

A-’Cf.R’. 6, x) o!xG(Alp-ep) /A (E,) +&pp (x)

x

in the case of Hilbert Yl . The relations (1.4) and (1.7) yield the inequality (1.5), while (1.4) and (1.8) yield (1.6). This proves the theorem. Note 4. If we further require in Theorem 2 that the metric in X be almost convex, then the inequalities (1 .S) and (1.6) can be strengthened, by replacing J(p, 6) on the right-hand side by J(p, 26). From the integral inequality (1.4) we obtain: Corollary

1

If the space X is compact, I.c(x) < =, the space Y1 is Banach, and the function f is not summable to the p-th power (p > I), then, for any mapping R, and for 6 > 0,

I

A'(f,

R, 6, x)dx==.

x

proof:If

sIlf(x) ilPdx=00, a point x.+X, will exist such that

x

s

IIf(x) lip dz==J

VDO,

C&,6)

since otherwise we might have chosen, for every point x=X a neighbourhood U(x, 6 (x) ) , 6 (x) >O, such that

s

If(x) Ilp dxca, L:CX,O,.K)) from the resultant covering of the compacturn by open spheres we could then have chosen a subcovering, and would have arrived at a contradiction. We note further that, by Minkowski’s inequality (see [4] , p. 13.5)

[

j Ilfw-f(~i> u(x,,b) -lM4 IIp”pw(z*,

il’dz,] 6));

“L

[

J IIf CJ(%,,a)

IIPdx2] i/p

V. A. Vinokurov

10 if the point X,,E U (x,, 6), then

IV(&)-f(G) s u(x,,e)

II”&,=a.

Using Tonelli’s theorem to write the right-hand side of (1.4) as a repeated integral, we get

s

4P(f,R,6,x)dxz3---

1 2pd6)

x

J { j Ilf(x2) -f(xd x U(X,,d)

where the inner integral becomes infinite for all xrEu( XO, 6) ; since p hand side of the inequality becomes infinite; this proves the corollary.

II’a}

h,

(~(XO,

6) )

>O, the right-

2. Estimation of the error integral for differentiable functions We consider an open set G of Euclidean n-dimensional space En with Lebesgue measure /A,and the locally summable function f: G+E”. Let ga, &ez (0, 6,)) be an arbitrary single-parameter family of mappingswith domain of definition D (g,) 1G in E” and with domain of values in Em. We shall show that, for a function f which is differentiable in some sufficiently wide sense.

Es 0

4’(f, g,, 6, ~1 dx

i/P

1

~C(f)6fo(W,

where cy) and o(6) are independent of the family ga. We shall introduce some necessary notation. In the present section, we shall identify the matrixA= {au}, l
(2-l) Using the numerical inequality

(2.2) it is easily shown that k-‘r-“’

( p, i=*

p>l.

fl,lajj[P) “p<(Al
i=t

j=t

(2.3)

Integral error estimates

We shall next consider the matrix functions A(x), defined on the open set GcE”, introduce the notation

11 and we

where IL4IIS= llA IIin the case Q = G. It follows from (2.3) that the norm (2.4) is equivalent to the norm

.

[

f’,Iaij(z) Ipds]“‘.

J F, Q

L-1

j-1

If the components of the functionfi G + Em have all generalized first-order partial derivatives in the sense of [S] , then &/(x)/ax will be the mXn matrix (rj$/&+}, l
where o(6) is independent of ga, 6~ (0, do) : space WP(I) (G) is understood in the sense of [S] ; this sense is not in general the same as the sense defined in [6]. The

hoof: We need to show that, given any c > 0, there exists 6’> 0 such that, for 6~: (0,6’]

II; II-4.

IIA(f,gd,6,z)llXj

(2.5)

We therefore fix an arbitrary E > 0 and show that such a 6’exists. By hypothesis, Ildf/i3sll -C00, so that there exists (see [7], p. 150) a continuous mXn matrix function $(x) with compact support KcG such that

Ildf/dx-cpll~&l8.

,

(2.6)

Further, since K is a compacturn and Kll (E”\G) =A, we have p(K, E”\G) =c>O. Consequently, for any point x of the generalized open sphere V= U (K, c/2) cG, U (5, c/2) cG. = c/16; then we have Weput U( v, 86) CG, We shall state a lemma, to be proved later.

l&(0, &I.

(2.7)

12

V. A. Vinokurov

Lemma 1 If V is an open bounded set and the rXk matrix function cp4’ (v) then, given any E > 0, there exists 62 > 0 such that for all 6~ (0, &] open sets Qs’czV, l<.%N (E), exist, along with a kX 1 matrix h(x), defined on the set

W(E) U Qld=Q’ and constant on any set Qs&,while

] h(x) 1G 1 Vz=Q*, p (Qf”, QP) 286, i+i,

Ikphll~~> IIcpII-48.

(2.8)

We can assume that 62 G 6 1. Since, given the point s=Q” for 6~ (0, 6J we have I/‘( z, 26) cG, the fundamental inequality for the convex case (see [2], Sec. 3) may be used for the points x:EQ’ and (a+@(z) ), ye (0, 2), i.e.,

We raise this inequality to the p-th power and apply the inequality (2.2); we get

Since the function h(x) is continuous in the set @, the functions A (f, g,, 6, s-i+@(s) wi11remain measurable functions of x on @, and we can integrate (2.9): and f(s+ydh(s))

s

Ap (f, g,, 6,x) dx +

Qb

)

s

Ap (f, g,, 6, xfy6l.s (x) ) dx

96

(2.10) 1

q-J Since

QacVcG,

we

If(x+y6h(x))-f(~)I*~~. Qb

have (2.11)

On the other hand, by Lemma 1,

s

A=(f, gb, 6, x+ydh (5) ) dx =

Qb

N

cs

AP(f, ge,6, x+yW

a=1

dx

Q.6

(2.12)

It remains to note that, in accordance with (2.8), the sets Qja+y6h,, are disjoint, and Q:+ySh,cG, s=l, 2,. . . , N, so that

13

Integral error estimates

N

s

Ap(f,gb, 6, x) da:2

G

ES s=l

A.”tf,

gb, 6, X)dX.

(2.13)

Qa8+T6h,

The inequalities (2.10)-(2.13) lead to the result

Ap(f,

gb, 6, X> dX2

$ J If(xt-y6h(x))-f(x)

IPdx, S=(O,&],

Qb

G

or

IIA tf,

‘+Iif (x+lbh (5)) -f

gb, 6, X) II>

SUP Ilf (x+@(X))

The quantity

-f

(5)

(2) /IQ', 6E(0,621,

(2.14)

7~(0,22).

IIQs is independent of the family g,, 6~ (0,

60))

so that the theoremYw6,’gk proved if we can show that

SinceU(x,86) cG for any point x=Qb, we can assume, in view of the continuous dependence of the left-hand side of the inequalities (2.8) on h, and also Taylor’s formula of [8], that, for almost all X-Q” we have i af ~-&x+tySh(x))h(x)dt.

f(xS-ybh(x))-f(x)=@

For the derivative af (z+tybh g

(x)

(x+tybh(x))-q(x)=

) /dxwe

(2.16)

write the identity

[$ (x+trsh(x))-~(x+tysh(x))]

we recall that the norm of an integral is not greater than the integral of the norm (see [7] , p. 166):

1j;

(xSty*h(x))h(x)dt-

0

j~(x)h(x)dt 1q 12 (5 J

”

+ty6h(x))-cp(x~tyGF,(x))

1dt +I(I~(x+tlSh(x))-~(s) I&, 0

and we obtain

i-af(xitydh(x))+)d~~~ ~~~~(~)~(~) IIS ”dx iiQ6

0"

-

II/

0

(~(x+tySh(x))--cp(z+t7sh(z))

IdtllQb

(2.17)

14

V. A. Vinokurov

for?= (0,2),6~(0,&). By Minkowski’s inequality (see [S] , p. 88), we have

(2.18)

11J lcp(~+whw-cpb) h-q *=q Il~(s+trsFY(s))-(p(z)llp8cEt. 0

(2.19)

0

But, by Lemma 1 and the inequality (2.6), we get

a-l

-cp(x+~yah,)

1‘cix]I”=[ 2 s=i

QP

12 (x> -cp(x) ’‘cix’ “’ (2*20)

J Cs)+fT6hr

G

We have again used the fact that the sets Q.8-l-tyGh,cG different subscripts s.

and that these sets are disjoint for

Since the function $(x) is uniformly continuous and has a compact support, we can choose 6&&

such that Icp(s+y)-q(s)

for 1y1<2&,

I.< 8(p($)i,p

and then

(2.2 1)

We have from the inequalities (2.18)-(2.21):

andbyLemma1, Il~(x)h(x)llQd~IlrpllV-~=

lk&-$. .8

But, recalling (2.6) and (2.16), we arrive at the inequality

Integral error estimates

15

which proves (2.15), and hence the theorem. We shah now give the proof of Lemma 1. Let VG?$” be an open bounded set, let 7 be its closure, and let the rXk matrix function $J(x) be continuous in v. Since @(x)is continuous, there exists, for any point x,,=V an open neighbourhood of the point Uxo in En such that

By definition of the norm, a kX 1 matrix h(xo) exists, 1h (50) 14, that, for xEU,O

1cp(20) h(G)

I= 1rp(50) I ; SO

hJ(4~2~00 I=cp(~o)~(~o)+(cp(x)-(P(xoca))~(x0) I (2.22) >Iq(x)l-

u 2(lr.(V)P

Since the set vis compact, we can isolate, from its covering by open sets of the type U,, a finite subcovering U,,, s=l, 2, . . . , N. For every open set L,= Vfl U,, we define the open set Wsa= (s6TY p(x, E”\L,) >96} and the closed set P:= (asEn: p(x, E”\L,) 26). Obviously, W.bd’,aG,~V.

We define the sets s=l,2

Q,a=WB6\,UFib,

1
,...,

Q”-

N,

; Qa6, a=,

and the function h (2) =h (3,) =h, fix XEQ,’.and we show that 62 > 0 can be chosen in such a way that

O'\Qa)<&,

A = suplq(x) xe7

I,

6E (0, 621.

(2.23)

In fact, if x=V\Qa and s is the minimum number of a set Lb which contains x, then XC?U Li i
V\Q”c

6 (L,\W).

Since, for every s, lim p (L,\ b+o

W,‘) =O,

(2.23) is proved. Let USfind a lower bound for

ITb)h(X) I+(x)

lltp(5) h(s) llg for 6~ (0, 621.From(2.22),

I-

2(p(;))i,p 9

==Q',

(2.24)

16

V.A. Vinokurov

Let x(x) be the characteristic function of the set Qs; we have (2.25) But, according to (2.23) (2.26) Combining the inequalities (2.24)-(2.26),

we obtain

Ilcp~llQs’>ll~ll”-~, 6Eo4 621, which proves Lemma 1. To obtain upper integral error estimates we shall require the following theorem, proved in [8] : Theorem 4 If the function f=W$ eitherpl>n,orp=l,l=n,then

(G) and can be continued up to the function ji~w(r? (E”) , where !]Y(z,

Y (x,6) =

6)]],=o(Sr)

as6+O,where

SUP YSGfl~(~,6)

The required upper bound for the error follows easily from Theorem 4. Theorem 5 If the functionfEW:”

(G)fl WA”(G)and can be continued up to the function fiEvi’)

(E”) fl IV;”(E”) ,wh ere either pl > n, or p = 1, I = n, then, given any reasonable algorithm of approximate solution Ra: G+E”, 6~ (0, &) , we have

llA(~,ll,,6,s)ll~26(1~11+r(26)

as

6-4

while in the case Ra=f VbTGs (0, &,)

iiA(f,f,6,s)ll~~1(~)(+r(6)

as

where y(6) is independent of the family Rat 7 (6) =O (6) for bi,

6-0, 7 (8) a-0 (8’) for 122.

Proof. Corollary 2 of [2] holds for a reasonable algorithm, i.e., By Theorem 4, o(f,hz)=

SUP

mk-fw

Y~WlU(X,~)

I<

1:

A (f, Ra, 6, 5) Qo (f, 26,~).

I6fqk

s>,

whereI]q(Z, 6)ll=‘f(6)* It remains to note that A (f, f, 6, 5) =a (f, 6, a~),and Theorem 5 is proved.

Integral error estimutes

17

Let us obtain a lower bound for the measure of the set in which the error is large. Theorem6 Let p(G) < = and for any function g: G + Em, let IIA (f, g, 6, x) ]IZL (f, 6) for a given S >O;then, A (f, g, 6, z) >eby_t(G),

p{x=G:

~(f,~)PhdW-~P Co(f, 26) *--eP

.

(2.27)

Proofi Using Theorem 1 of [2], we pass from the mapping g to the &reasonable mapping v: G -+Em such that A (f, u, 6, x) GA(f, g, 6, x), XEG, A (f, U, 6, x)a}, we estimate the error integral as follows:

JA*(!, u, 6, x)dxf J op(f,2s)dx+~P[~F1G)-~(EE)l

L(f,6)*< +(E,)

[o’c(f,26)-EP]+ep~(~),

whence (2.27) follows. From Theorems 3 and 6 we get:

If p(G) < = and the function f satisfies a Lipschitz condition with constant K, then p{x=G: as

A(f,Rd,x)>~~~~~(G)

6+0

for any family of mappings Ra: G+E”, and E.

llaf/axll’/~(G)-e”+o(l) (2K) p-ep

6~ (0, 6,,), where o( 1) is independent of the family Rg

The meaning of the inequality (2.28) is most clearly seen in the case p=m=n=l,

as

(2.28)

G= (0, 1)

S+O. 3. Inverse problems. Probabilisticinterpretation

The technique of solving unstable inverse problems gave rise to the problem of regularization, so that the following situation is of interest. Assume that the measure I-(is given in the space X, along with the one-one E.c-measurablefunction f: X2 Y, but assume that the approximate solution is sought, not for the functionf, but for the function fr 1; here, given the isometric embeddings Y,~Yand X,1X and the mapping R from Yl into X,, D(R) ‘Y, we define the error of the solution of the converse problem at the point x=X as Ao(f, R, 6, x) =A (f-‘, R, 6, f(x)),

(3.1)

18

V. A. Vinokurov

i.e., the error is linked, not with the point ytY, but with the point x =f-l(y). extend the integral inequality (1.3) to this case.

Our aim will be to

Recall that we define an analytic set (see [9], pp. 464,489) as a continuous image of a B-measurable subset of a complete separable space. Lemma 2 If X is an analytic set with finite regular non-negative measure, with respect to which all open sets are measurable, and f: X2 Y is a p-measurable one-one mapping, then the image of the measure p in the space Y is a measure with the property S. Proo$ The image v of the measure p under the mapping f is obviously a perfect non-negative finite measure; since fis p-measurable, every open set of the space Y is v-measurable. We only need to show that every u-measurable subset of the space Y is congruent to a set of the type Fo. Since the measure /Ais regular, a type F. set EcX of perfect measure exists, such that the contraction jl~ is a B-measurable function of the first class (see [3 ] ). Let us show that the contraction of the measure v onto f(E) is a regular measure. But, since the measure ~1is regular, the contraction PIE is also regular, i.e., given any p-measurable set A cE a type F, set K exists in E such that KG4, p((A\K) =o, i.e., v (f(A) \f (K) ) =O. By the inverse function theorem, the inverse mapping for the B-measurable function jls, given on the analytic set E, is B-measurable, (see [9] , p. 500), i.e., f(K) is a set which is B-measurable in f(E). We have shown that, given any u-measurable set f(A) cf(E) there exists a subset of equal measure contained in it which is B-measurable in&F). Notice next that, if we consider the measure v 1 f(E) only on the completion of B-measurable subsets of the set A&‘),this will be a regular measure (see [lo] , pp. 26-27), i.e., for every v-measurable subset f(K) cf(E) there exists a set L of type F, in f(E), kf(K), of the same measure. In short, v 1fp) is a regular measure, but every v-measurable set LcY v-measurable set L fl f (E) , L-LflJ(E)

is congruent to a

(mod v),

which, by what has been proved, is congruent to a set Tcj( E) of type FG in f(E) , Lflf(E)-T

(modv),

while, since the topology in fi&‘) is hereditary, the set T has the form T=T,flf(E), where T1 is a set of type F, in Y. We obtain L-Lnf(E)-T-T,

(mod v),

i.e., the lemma is proved. We are now able to extend the integral inequality (1.3) to inverse functions. nteorem 7 Let X be an analytic set wit& perfect non-negative measure P, with respect to which all open sets are measurable, while X= ( U M,,) UP, where M, are sets of finite measure closed in X, in which n=i

Integral

error

19

estimates

the contraction of the measure is kregular, while cc(P)= 0; let f be a pmeasurable one-one mapping f: XZY; let X,zXand Y,zY be metric extensions, and R a mapping from Yl intox,, Then the functional A0(f, R, 6, z) is a pmeasurable function of x in X and we have

D(R)1Y.

or if

then s

R,&x)dx2

Ao”(f,

X

p%,

s x

G)dX2

x*EX:P(f(r*),f(x,))C~

I

dx,

pa.

&of. By Lemma 2, the image of the measure /J is a measure with the property S in the set R&r); but then, by Theorem 8 of [2], A (f-l, R, 6,y) is a u-measurable function ofy in f(M,,) cY, but this implies, in view of the equation Ao(f, R,6,x)=A (f-l, R,6,f(x) ) and the definition of the measure YinR&r), that the functional A0(f, R,6,x)is pmeasurable in M,, .

R, 6,x)isthus /f-measurable in the union ( : M,)

The functionalA,,(f,

n-i

UP=X.

Since the measure v =fi) has the property S in the set fTMn), Theorem 1 holds for the u-measurable function f -l(M,), i.e.,

Ap(f-‘, R, 6,ydv {wf WJ : p (i/i, YZ>-+%‘i J f(Mn) 1 app (f-’h/i)t f-’(!A 1&ii dYz* SJ 2p Y,,YI~f(~n):P(IIL.u1)~~ By deftition

of the measure V,this inequality can be written in the variable x: s

Ao”(f, R, 6, G> ~{w=&:

XII

p (f

(4, fbz> ) <+k

1

a~2p

ss .I,~z~~,:P(I(r‘).f(Y

pp (Xi,

x2) dXi

dxz.

It can be assumed without loss of generality that M,cM,+I, n=l, theorem on monotonic convergence to a limit (see e.g., [4], p. 168),

2,. . . , so that, by the .

AoP(f,R,6,xi)p{w=X: p(f(xi), f(~z))<~~~~l J .. =lim

n*ol

J AoP(f,R,6,x!)Cl{xz~~,: XII

p(f(dyf(d)<6)dX~

20

V. A. Vinokurov

and X,,~ZEX

JJ

pP(x,, x,) dx, dxn

P(f(z,),f(xz)i<6

JJ

= lim

n-tm

whence the theorem follows. Note 5. If, in Theorem 7, we have the supplementary condition that the metric in Y be almost convex, then the inequality in the theorem can be strengthened (see Note 3), by replacing p (f(s,), f(d)4 by ~(f(x,),f(z,))c26throughout. The conditions of Theorem 7 are satisfied by Radon’s measure on a locally compact space, countable at infinity, or on an analytic subset of the space, or by Wiener’s measure on L,[0,11or Co[O, 11, etc. A probability measure on a space X can be interpreted in such a way that, prior to obtaining approximate input data, the possible values sr-X are expected with certain probabilities. Hence, if we are given a method of approximate solution, we have an 12priori distribution of the error probability at a point for the method. Consider the space X with a probability measure having the property S; then Theorem 8 of [2], on the measurability of the error functional for any approximate mapping, enables the Theorem 8 of [3] to be strengthened in the following way (see in [3] for the deftition of “regularizability almost surely”). ?heorem 8

Any random functionf, defined on a metric, but not necessarily separable, space X, with a probability measure having the property S, and taking values in separable metric space Y, is regularizable almost surely. hoof. By Theorem 6 of [3], there exists a family of mappings

Ra:X+Y, O-G=Qo,such

that lim A(f,

Rb,6,z)=0

b+O

for almost all x. Since, by Theorem 8 of [2], L!I(f, &, 6, 2 ) is a measurable function of x, this proves the regularizability almost surely. In the case of a probability measure, Theorems 1 and 7 yield lower bounds for the mathematical expectation of the error and for its moments of any order, these bounds being independent of the method of solution; and they lead naturally to the question of constructing methods of approximate solution giving the least mean error, i.e., to quantitive criteria for the optimality of a method. Investigations along these lines should be of great interest for the theory of approximate methods. 7kanslated by D. E. Brown

21

Solving a classof operator equations

REFERENCES 1.

VINOKUROV, V. A., Asymptotic error estimates, III, Zh. v?chisl. Mat. mat. Fiz., 16, No. 1, 3-19, 1976.

2.

VINOKUROV, V. A., Properties of the error functional AC R, 6, x) for fixed 6 as a function of x, I, Zh. vychisl.Mat. mat. Fiz., 15, No. 4, 815-829, 1975.

3.

VINOKUROV, V. A., Regularizability almost everywhere, Zh. vychisl.Mat. mat. Fiz., 14 No. 3, 560-571, 1974.

4.

DUNFORD, N., and SCHWARTZ, J. T., Linear operators, Vol. 1, Wiley, 1958.

5.

SOBOLEV, S. L., Introduction to the theory of cubature formulas (Vvedenie v teoriyu kubaturnykh formul), Nauka, Moscow, 1974.

6.

NIKOL’SKII, S. M., Approximation of functions of several variablesand embedding theorems (Priblizhenie funktsii mnogikh. peremennykh i teoremy vlozheniya), Nauka, Moscow, 1969.

I.

BOURBAKI, N., Integration. Measures, integration of measures, Hermann, Paris, 1952.

8.

VINOKUROV, V. A., On the differentiability No. 1, 15-18, 1976.

9.

KURATOWSKI, K., Topologie, Hafner, 1958.

of functions of Sobolev space, Dokl. Akad. Nauk SSSR, 227,

10. PARTHASARATHY, K. R., Probability measures on metric spaces, Academic Press, 1967.

APPROXIMATE METHODS FOR SOLVING A CLASS OF NON-LINEAR OPERATOR EQUATIONS* A. G. ZARUBIN and M. F. TIUNCHIK Khabarovsk (Received 15 July 1974) THE CONVERGENCE of the approximate solutions for a class of non-linear operator equations is discussed. Schemes of investigation are described which embrace Gale&in’s method, the method of moments, and difference methods. 1. Let E and F be Banach spaces over the real number field. Assume that two operators A and K, acting from E into F, are given. We assume that I?& (E, F) , where L(E, fl denotes the space of linear continuous operators, acting from E into F and defined in the whole of E. The operator K is non-linear. We consider the equation .AuSKu=f,

(1.1)

where u is the required element of E, and fis a given element of the space F. Operators acting from E into F will be approximated by a sequence of operators, acting from certain other Banach spaces En into Banach spaces F,, n- 1, 2, . . . The spaces E and Fare connected with the spaces En and F,, by the mappings p,,c&(E, ER) and q,,EL(F, F,) with the properties:

*Zh. vfchisl. Mat. mat. Fiz., 16,3, 567-576,

1976.